I would recommend exim4. It's configuration is probably the easiest from the classic trio of sendmail, postfix and exim. Also leaves small footprint. We use it on our webservers for exactly the same purpose

It's the default mail server, so you'll have all the distribution-specific troubleshooting guides you might need;

It sends mail trough secure connections.

The functions you don't need won't affect the ones you need anyway, and it's not like one megabyte of disk space is a problem - at least in a context where it's okay to install a behemoth sugh as Ubuntu.

I use sSMTP to relay mail through a Google Apps e-mail account on a Gentoo-based Linode VPS running Apache with mod_php. Since it's an outbound only sendmail replacement that only relays to other servers for them to actually deliver the mail, this sounds like the ideal request.